Consider the evolutionary process under fixed environmental conditions, where genetic change leads to phenotypic change, and fitness is given as a function of phenotype. In this case, the variance Vip of the fluctuation of the phenotype due to noise is proportional to the rate of evolution of the phenotype, termed as evolutionary fluctuation–response relationship. It then implies that Vip is proportional to Vg, the variance due to genetic variation, as derived theoretically under the assumption of evolutionary robustness: the acquisition of phenotypic robustness to noise through evolution also leads to robustness to genetic variation. Here, as the mutation rate increases (or the noise level in the dynamics decreases), a phenotypic error catastrophe occurs, where it is no longer possible to maintain the fit phenotype. While phenotypic variance and evolvability decrease under fixed environmental and fitness conditions, they rise and fall repeatedly as environmental conditions are varied over generations. Phenotypic plasticity and evolvability are maintained under environmental variation. Strong selection under fixed evolutionary conditions can lead to the appearance of mutants with increased phenotypic variance. This may be due to over-optimization to obtain the fit phenotype, which may break consistency with other processes and reduce robustness.

In this chapter, we explore theoretical aspects of the origin of life problem. Firstly, we address the Chicken and Egg problem referring to the “RNA world.” We explain a mathematical model of the RNA replication system introduced by Eigen and discuss the conditions necessary for self-replication, referring “error catastrophe.” As a potential solution, we discuss the “hypercycle,” alongside its vulnerabilities and the acquisition of evolvability through compartmentalization. On another front, we examine Dyson’s catalytic reaction system as an alternative hypothesis, showing that catalytic reaction networks capable of maintaining themselves and undergoing imperfect reproduction may have appeared first. We also refer to a simple model of polymer reactions, arguing that such autocatalytic reaction networks can stochastically emerge, as proposed by Kauffman. Furthermore, we describe a cell model featuring an intracellular chemical reaction network that divides based on its state, highlighting the universal nature of reaction dynamics in replicating cells and the power-law distribution of chemical abundance (Zipf’s law), which has been verified across many organisms. Additionally, we introduce the concept of “minority control” in catalytic reaction networks, which can carry primitive genetic information. Finally, we discuss perspectives on research regarding the origin of life.